This document is published in:

International Journal of Impact Engineering

, 2013, 54, 206-

216. Doi:

http://dx.doi.org/10.1016/j.ijimpeng.2012.11.003

.

© Elsevier

Finite element analysis of AISI 304 steel sheets subjected to dynamic tension:The

effects of martensitic transformation and plastic strain development on ﬂow

localization

J.A.Rodríguez-Martínez

a

,

*

,D.Rittel

a

,

b

,R.Zaera

a

,S.Osovski

b

a

Department of Continuum Mechanics and Structural Analysis,University Carlos III of Madrid,Avda.de la Universidad,30,28911 Leganés,Madrid,Spain

b

Faculty of Mechanical Engineering,Technion,32000 Haifa,Israel

o

Keywords:

Dynamic tension

Nec

king

Strain induced martensitic transformation

Critical impact velocity

a b s t r a c t

The paper presents a ﬁnite element study of the dynamic necking formation and energy absorption in

AISI 304 steel sheets.The analysis emphasizes the effects of strain induced martensitic transformation

(SIMT) and plastic strain development on ﬂowlocalization and sample ductility.The material behavior is

described by a constitutive model proposed by the authors which includes the SIMT at high strain rates.

The process of martensitic transformation is alternatively switched on and off in the simulations in order

to highlight its effect on the necking inception.Two different initial conditions have been applied:

specimen at rest which is representative of a regular dynamic tensile test,and specimen with

a prescribed initial velocity ﬁeld in the gauge which minimizes longitudinal plastic wave propagation in

the tensile specimen.Plastic waves are found to be responsible for a shift in the neck location,may also

mask the actual constitutive performance of the material,hiding the expected increase in ductility and

energy absorption linked to the improved strain hardening effect of martensitic transformation.On the

contrary,initializing the velocity ﬁeld leads to a symmetric necking pattern of the kind described in

theoretical works,which reveals the actual material behavior.Finally the analysis shows that in absence

of plastic waves,and under high loading rates,the SIMT may not further increase the material ductility.

2012 Elsevier Ltd.All rights reserved.

1.Introduction

The characterization of the mechanical properties of materials

at high strain rates has become increasingly relevant for the

industry.Accurate knowledge of those properties is usually

required in different engineering applications such as aeronautical

[1,2],automotive [3,4],naval [5,6] and manufacturing [7,8],where

service conditions involve large strains at high strain rates.

Among the experimental characterization tests,the uniaxial

tensile arrangement is deﬁnitely the most commonly used due to

its simplicity.The uniformstate of stress and strain along the gauge

for this test greatly simpliﬁes the interpretation of results.However

the homogeneous deformation eventually ceases due to the onset

of necking in the specimen.Necking in uniaxial stress conditions

has been largely studied since it indicates the onset of a process

which leads to material failure,and therefore determines the

suitability of materials to absorb energy.For quasi-static loading,

Considère [9] showed that in a long and thin bar,the neck develops

at maximumload.Within the framework developed by Hill [10] on

the theory of bifurcation in elasticeplastic solids,different authors

[11,12] demonstrated that the strain at maximum load provides

a lower bound to the strain at which bifurcation occurs;in speci-

mens showing large length/width ratio localization starts slightly

after the maximum load whereas in specimens showing small

length/width ratio localization is further delayed.

Under dynamic conditions,the onset of necking is additionally

inﬂuenced by other factors.Different authors concluded that the

condition for instability in tensile tests also depends on the strain

rate sensitivity of the material;Woodford [13] and Ghosh [14]

collected data from tensile tests of a number of metals,showing

a strong delay in necking with increasing logarithmic strain rate

sensitivity.In the meanwhile,a number of theoretical works were

developed to explain this inﬂuence.Hart [15] formulated a stability

criterion for materials exhibiting strain rate sensitivity that was

later re-examined by Ghosh [16].Klepaczko [17] developed

a theoretical framework to include the effect of temperature in the

analysis of instabilities in rate-dependent materials.Hutchinson

and Neale [18] concluded that strain rate sensitivity has a strong

* Corresponding author.Tel.:þ34 916248460;fax:þ34 916249430.

E-mail address:jarmarti@ing.uc3m.es (J.A.Rodríguez-Martínez).

inﬂuence on the post-uniform elongation,retarding necking

localization.In the recent years,a number of papers has been

published which provided further understanding of the interplay

between strain rate sensitivity and necking formation [19e23].

Inertia was the following effect considered as inﬂuential in the

development of dynamic instabilities.Fressengeas and Molinari

[24] used a perturbation analysis to discuss dynamic effects on

ductility,showing that geometrical perturbations are stabilized by

inertia effects.Using a bifurcation analysis,in this case under plain

strain conditions,Shenoy and Freund [25] demonstrated also the

stabilizing effect of inertia.More recently,one should mention

a number of theoretical and numerical works that have provided

additional veriﬁcation of the beneﬁts provided by material inertia

to delay necking formation [19,20,26e29].Material strain hard-

ening is certainly an additional factor which enhances ductility.The

Considère condition [9] clearly outlines the favorable effect of

a high strain-hardening coefﬁcient in quasi-static conditions,but

this effect has also been addressed by different authors under high

rate loading conditions [16,17,21,29,30].These studies considered

the inﬂuence of parameters,such as material inertia,strain hard-

ening and strain rate sensitivity in absence of wave propagation

phenomena.

Among the previous effects modifying the onset of dynamic

necking,strain hardening is known to be inﬂuenced by micro-

structural effects such as dynamic phase transformations.Speciﬁ-

cally,Strain Induced Martensitic Transformation (SIMT) acts as

a highly potent mechanism of martensite germination associated

with plastic deformation in the austenitic phase.The trans-

formation of austenite into martensite is comparable to a dynamic

composite effect due to the progressive appearance of the

martensite during straining,enhancing strain hardening of the

steel.Different authors have recently addressed the role played by

the SIMT in the dynamic behavior of different steel grades [31e34].

It was observed that this phase transformation mechanism can

reasonably be expected to affect the propensity of a material for

dynamic necking,a point that deserves further investigation.

High-speed hydraulic machines and Split Hopkinson (Kolsky)

Bar devices have been successfully applied to investigate the

mechanical behavior of materials at intermediate to high strain

rates [35,36],the goal being the determination of the intrinsic

properties of the tested material under uniform uniaxial stress

conditions.Therefore different authors introduced original tensile

specimens to optimize the geometry,favor a homogeneous strain

ﬁeld and delay instabilities [37e40].However,imposing a velocity

boundary condition on one side of a solid at rest ewhich is needed

for dynamic testing e necessarily produces a plastic wave front.In

such a case the strain gradients due to the effect of plastic wave

propagation in the specimen may hinder the actual constitutive

behavior of the material.Under large impact velocities,the strain

ﬁeld becomes rapidly non-uniform,affecting the ductility of the

specimen and shifting the position of the neck in the sample.

Hopkinson [41,42] and Hopkinson [43] analyzed the dynamic

loading

of steel

wires and observed that they broke at different

points along their length depending on the loading velocity;the

elastic wave propagation phenomena were considered as respon-

sible for this behavior.Furthermore,Von Kàrman and Duwez [44]

and Clark and Wood [45] reported a Critical Impact Velocity (CIV)

such that,when exceeded,the force equilibriumis not fulﬁlled and

necking occurs close to the impacted end with negligible subse-

quent plastic strain in the rest of the specimen.The ﬁrst theoretical

work on CIV was proposed by Von Kàrman and Duwez [44],while

Klepaczko [17,46] extended this theory to consider strain rate and

temperature effects.

In order to avoid the drawback of the wave disturbances on

determining material ductility at high strain rates,the ring

expansion test was developed [47] and investigated by different

authors [19,30,48,49].Here,complications resulting from wave

propagation are eliminated until the onset of necking due to the

symmetry of the problem,which implies that the inﬂuence of

loading velocity on material ductility can be studied,virtually,

without limits on the applied velocity.However,the complicated

experimental arrangement required for the ring expansion test

impedes the determination of the material stress-strain charac-

teristics.Thus,the uniaxial tensile test,in its dynamic version,

cannot simply be ruled out because of suspected wave-related

effects,at the beneﬁt of the ring expansion test.This remark calls

for an in-depth additional evaluation of the dynamic tensile test,in

order to better assess its beneﬁts and also its limitations,while

taking into account the occurrence of dynamic phase trans-

formation of the above-mentioned kind.

This paper investigates necking formation in AISI 304 steel

sheets subjected to dynamic tension.This steel grade is considered

a reference metastable austenitic stainless steel for studying the

SIMT process at high strain rates since it shows a large amount of

transformed martensite even under adiabatic conditions [50].The

analysis emphasizes the effects of martensitic transformation and

plastic wave propagation on ﬂowlocalization and sample ductility.

For that purpose,ﬁnite element simulations of a dynamic tensile

test have been performed,in which the SIMT has been switched on

and off to disclose the effect of the enhanced strain hardening

produced by the SIMT.In addition,two different initial conditions

have been considered:specimen at rest and specimen with an

initial velocity ﬁeld in the gauge.The analysis shows that inabsence

of an initial velocity ﬁeld and within certain ranges of impact

velocity the neck may be shifted fromthe middle toward the ends

of the specimen.The shifting of the neck is found to be a limiting

factor for the sample ductility.On the contrary,the initial velocity

ﬁeld minimizes the propagation of plastic waves along the longi-

tudinal direction of the sample,thus leading to a rather symmetric

necking pattern of the kind described in aforementioned theoret-

ical works [19,27].Plastic wave propagation phenomena not only

affects the ductility of the specimen but may also invert the ex-

pected increase in energy absorption due to the SIMT.Furthermore,

the role played by the plastic wave propagation hiding the actual

material behavior has been highlighted;and a range of strain rates

for which the measured dynamic tensile characteristics of a mate-

rial can be considered as actual material properties has been

determined.Finally the analysis shows that in absence of plastic

waves,and under high loading rates,the SIMT may not provide

further beneﬁts to the material ductility.

The paper is organized as follows.Section 2 provides a brief

summary of the thermo-viscoplastic constitutive equations used to

model the mechanical behavior of the AISI 304 steel and empha-

sizes the enhanced strain hardening of the material due to

martensitic transformation.Section 3 describes the different ﬁnite

element models developed to perform the study.In Section 4 the

results of the numerical simulations are shown and discussed,

focusing the attention on the effects of martensitic transformation

and plastic wave propagation on ﬂow localization and sample

ductility.The concluding section outlines the main outcomes of this

study.

2.Constitutive modeling of strain induced martensitic

transformation and calculation of effective properties

A complete description of the model,the values of the param-

eters identiﬁed for AISI 304 stainless steel and its validation with

dynamic tensile tests results can be found in Zaera et al.[34,51],but

here the key points of the constitutive formulation are further

discussed for completeness.The constitutive description is based

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216 207

on the previous works of Olson and Cohen [52],Stringfellow et al.

[53] and Papatriantaﬁllou et al.[54],all developed to account for

SIMT in steels containing metastable austenite,and includes

modiﬁcations in the following items:

e According to Olson and Cohen [52] the intersection of shear

bands in austenite is considered as the dominant mechanismof

SIMT.The kinetics of the transformation is described by an

exponential expression in which the plastic deformation in

austenite is multiplied by a coefﬁcient

a

favoring the shear

band deformation mode.The parameter

a

decreases with

temperature,as observed by Olson and Cohen [52].The

following phenomenological law is proposed to ﬁt this

dependence

a

ð

Q

Þ ¼

a

0

h

1

Q

exp

a

1

1

Q

1

i

(1)

where

a

0

and

a

1

are constants and

Q

is a normalized temper-

ature.In comparison with the polynomial expressions applied

by other authors to describe the change of

a

with temperature

[52,55,56],the one proposed in this work facilitates to capture

the decrease of the transformation rate with increasing

temperature within the temperature range in which the SIMT

occurs.

e The linearly temperature dependent normalized thermody-

namic driving force for the martensitic transformation,

proposed by Stringfellow et al.[53],is replaced here by an

exponential equation.This new law provides a greater

temperature sensitivity of the rate of martensitic trans-

formation.This becomes relevant at high strain rates where the

temperature rise of the material results from the adiabatic

character of the plastic deformation.According to the experi-

mental results reported by Rodríguez-Martínez et al.[50] at

high deformation rates,and therefore signiﬁcant temperature

increase,small temperature variations lead to signiﬁcant

differences on the volume fraction of martensite formed.

e The thermal deformation tensor rate is included in the gener-

alized Hooke’s lawfor hypoelastic-plastic materials,taking into

account that the adiabatic increase of temperature may lead to

non-negligible thermal strains.The rate of thermal deforma-

tion tensor is assumed to be isotropic

d

q

¼

a

q

_

q

1 (2)

a

q

being the coefﬁcient of thermal expansion,

q

the absolute

temperature and 1 the second order unit tensor.Mahnken et al.

[57] used a rule of mixtures to calculate

a

q

taking into account

the different coefﬁcients for thermal expansion of martensite

and austenite.However,different authors [58] shown that the

thermal expansioncoefﬁcient does not obeythelawof mixtures

because a state of microstresses appears between the phases

when submitted to a temperature increase.Therefore the

authors have chosen a constant and characteristic value of

a

q

.

e The thermal softening of each solid phase is considered in the

model through a thermo-viscoplastic potential,which has been

deﬁned in such a way that the yield stress follows a power

thermo-viscoplastic law.These laws are commonly accepted to

model strain/strain rate hardening and temperature softening

[19,59e61].The effective properties of the thermo-viscoplastic

heterogeneous material are calculated using a plastic potential

including viscous and thermal effects for both austenite and

martensite,using the modiﬁed secant method proposed by

Suquet [62e64] and the solution algorithm developed by

Papatriantaﬁllou et al.[54].

e Anoriginal thermodynamic scheme to capture the variability of

the TayloreQuinney coefﬁcient in austenitic steels showing

strain induced martensitic transformation at high strain rates is

included.Different heat sources involved in the temperature

increase of the material are taken into account.These are the

latent heat released due to the exothermic character of the

transformation and the heat released due to austenite and

martensite straining.The intrinsic dissipation in each of

the constitutive phases is computed as a constant fraction of

the corresponding plastic work power.The variability of the

TayloreQuinney coefﬁcient stems from the evolution of the

phase fractions and from the latent heat released upon phase

transformation.Through a differential treatment of these

dissipative terms,the TayloreQuinney coefﬁcient develops

a direct connection with the martensitic transformation,and

the proposed model strives to provide complementary insight

regarding heat generation in dynamically phase transforming

alloys,in which the heat power might be greater than the

intrinsic dissipation power.

The so-called dynamic composite effect due to the progressive

appearance of martensite is clearly shown in Fig.1.The yield

strength of martensite is higher than that of austenite,increasing

the ﬂow stress and strain hardening in the steel grade.

3.Finite element modeling of dynamic tension

Dynamic tension FE simulations are conducted in order to

evaluate the effects of SIMT and plastic strain development on the

energy absorption capabilities of the material.The simulations are

run for strain rates ranging between 500 s

1

_

3

20;000 s

1

.The

authors are aware that this range of loading rates exceeds the

regular experimental capabilities,however this way of proceeding

is efﬁcient to analyze the effect of the aforementioned mechanical

factors on the strain localization process.Two different numerical

conﬁgurations are addressed:full sample description and single

element simulations.

3.1.Full sample description

The geometry of the tensile specimen is taken fromRodríguez-

Martínez et al.[50],Fig.2.The mesh consists of 10,941 eight-node

σ

ε

Fig.1.Comparison between numerical predictions of the stressestrain behavior for

AISI 304 grade,in which the martensitic transformation has been switched on and off

_

3

¼ 500 s

1

.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216208

tri-linear brick elements with reduced integration.The integral

viscoelastic approach available in ABAQUS/Explicit [65] has been

used to prevent hourglass deformation modes,with the scale factor

used for all hourglass stiffnesses being equal to one.According to

the considerations reported by Zukas and Scheffer [66],the gauge

of the specimen has been meshed using elements whose aspect

ratio was close to 1:1:1 (z1/3 1/3 1/3 mm

3

).Loading in the

specimen has been introduced through the following boundary

condition:an axial velocity V instantaneously applied on side A e

see Fig.2 e which remains constant throughout the entire process

[40,67].

Two different initial conditions have been applied:

e No-ﬁeld:The initial condition is V ¼ 0.The application of this

initial condition to a specimen initially at rest leads to the

propagation of a plastic wave front along the longitudinal axis

of the sample [21,68].This conﬁguration is representative of

a typical experimental setup.

e Field:The axial velocity along the length of the sample V

1

is

initialized as schematically described in Fig.3.

This initial condition avoids the abrupt motion of the sample

along the axial direction at t ¼ 0.Here should be noted that the

initial velocity along directions X

2

and X

3

is set to 0.The authors are

aware that this leads to acceleration of the sample along X

2

and X

3

directions at the beginning of loading.Moreover,let us note that

since the reference conﬁguration is stress-free at the onset of

loading the sample is subjected to an initial stress increase as

a consequence of the load application.However,it is reasonable to

assume that the distortion of the strain ﬁeld along the longitudinal

axis of the specimen is the dominant mechanism governing the

process of necking formation.The potential strain disturbances

along X

2

and X

3

directions and the stress increase at the beginning

Fig.2.Schematic representation of the 3D model.Mesh,dimensions (mm),boundary conditions and loading condition.

Fig.3.Schematic representation of the initial velocity ﬁeld applied to the sample in

the ﬁeld simulations.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216 209

of loading may play a secondary role in the neck inception.In

forthcoming sections of this paper the effectiveness of this initial

condition in minimizing the propagation of the plastic wave front is

strengthened.

It is worth noting that in order to mimic most of the existing

experimental tests,no initial numerical or geometrical imperfec-

tion has been assumed.A regular mesh was used to minimize

mesh-related imperfections,so that the necking instability would

develop froman initially smooth surface.This procedure has been

shown effective in reproducing the localization mechanics of the

dynamic tension problemas demonstrated by Rusinek et al.[40].

3.2.Single element conﬁguration

FE models with a single element having one quadrature point

allowefﬁcient integration of the constitutive models under partic-

ular loading conditions.The effect of a given specimen geometry is

discarded and the results yield the speciﬁc response of the consti-

tutive models.Stress-strain characteristics obtained from this

conﬁguration are used as a reference for evaluation of the results

obtained from the full sample simulations.One 1 1 1 mm

3

8-node tri-linear brick element with reduced integration was used

inthesimulations.Fig.4shows theloadingandboundaryconditions

imposedtothe element todeﬁne the tensile stress state.It shouldbe

noted that for a single element having one integration point the

generation of a plastic wave front is precluded.There is no upper

limit in velocity which impedes determination of the stress-strain

material characteristics,i.e.the critical impact velocity is avoided

in this approach.

4.Results and analysis

The reported stress-strain curves are obtained from the full

sample numerical simulations following the regular procedure

used in experimentation.The sample displacement (used to

determine the strain in the sample gauge during the simulations) is

recorded at the impacted side of the specimen and the force (used

to determine the stress) at the clamped end e see Fig.2.Moreover,

similarly to Xue et al.[21] and Rodríguez-Martínez et al.[69],the

localized necking strain

3

neck

e fromthis point on denoted simply

by necking strain e is determined by the condition du

1

/dt ¼ 0,

where u

1

is the longitudinal displacement in the X

1

direction

measured at a node beside the necked zone,in the direction toward

the clamped side,and t refers to time,Fig.5.Once the necking strain

is known,it can be used as the upper limit of integration of the

stressestrain curves for the determination of the energy absorbed

by the sample until strain localization occurs,E

s

.

4.1.The role of SIMT in presence of plastic waves

The role of SIMT on the energy absorption capability of the

material in presence of plastic waves is evaluated using no-ﬁeld

initial conditions.For that task,the transformation has been

systematically switched on and off.Fig.6 shows the necking strain

3

neck

and the energy absorbed by the sample E

s

as a function of the

loading rate _

3

for simulations in which the SIMT has been activated

and deactivated.Note that 10% is the percentage-value of the data

considered,which is represented by the error bars.

Consider ﬁrst the general behavior of

3

neck

and E

s

as a function of

the strain rate _

3

.For this,we will just refer to simulations in which

the SIMT has been switched on e solid symbols in Fig.6.It can be

observed that for strain rates up to w3000 s

1

the necking strain

and the energy absorbed by the sample slightly increase with

impact velocity.This comprises the range of loading rates for which

necking takes place in the middle of the sample,Fig.6.Then,within

Fig.4.Schematic representation of the single element model.Boundary and loading

conditions.

Fig.5.Schematic representation of necked zone showing the measurement point used

for determination of the localized necking condition.

ε

ε

ε

ε

Fig.6.Necking strain

3

neck

and energy absorbed by the sample E

s

as a function of the

loading rate _

3

for no-ﬁeld simulations in which the martensitic transformation has

been switched on and off.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216210

certain range of strainrates,upto 4500 s

1

approximately,

3

neck

and

E

s

slightly decrease with impact velocity.This behavior corresponds

to the loading rates for which the neck takes place closer to the

clamped side,Fig.7.Whenthe neck is inceptedclose tothe clamped

side,the shoulder of the specimen limits the development of plastic

deformation inducing large strain gradients responsible for

decreasing

3

neck

.At higher strain rates,the necking location starts

moving from the clamped side to the impacted end leading to

a continuous rise in

3

neck

and E

s

with impact velocity which extends

until attainment of the CIV at w150 m/s (which corresponds to

7500 s

1

),Fig.6.Then,the sample ductility continuously decreases

with loading rate.Here it should be noted that in order to observe

a drastic reduction in ductility beyond the CIV,it is necessary to

analyze specimens showing larger length/width and length/thick-

ness ratios,as pointed out by Von Kàrman and Duwez [44].It is

worth noting that identical relationships between

3

neck

,E

s

and _

3

are

observed in the case of deactivated SIMT;the only difference lies in

the loading velocity values for which the neck takes place in the

middle of the gauge,in the clamped side or in the impacted side,

Fig.7.In the case of no-SIMT,the shift of the neck (both variants,i.e.

necking closer to the clamped side,necking closer to the impacted

side) occurs at lower impact velocities due to lower material ﬂow

stress and strain hardening.The calculated results concerning the

general dependence of necking strain and energy absorbed by the

sample on impact velocity and necking location are all well docu-

mented in the literature [40,45,67].This clearly supports the idea

that necking location and necking strain are direct consequence of

test loading rate,material constitutive behavior and plastic wave

propagation (in addition to other factors related to the geometry

and dimensions of the specimen,material density) and,especially

at high strain rates,cannot be considered the result of a random

process governed by material or geometrical defects.

Next,let us identify the contribution of the SIMT to the calculated

values of

3

neck

andE

s

.At lowstrainrates,uptow4000s

1

,thenecking

strain and the energy absorbed by the sample until necking are

slightly larger if the SIMT is active.This behavior could be expected

since the martensitic transformation increases the material strain

hardening,the latter being a variable involved in necking retardation

[21].It is worth noting that the relationshipbetweenimproved strain

hardening and necking retardation has been frequently presented as

universal in the literature.However this is not the case for boundary

value problems involving plastic wave propagation.This is illustrated

inFig.6 for the range of strain rates 4000 s

1

< _

3

< 6000 s

1

.Within

these loading rates both necking strain and energy absorbed by the

sample are larger when the martensitic transformation is switched

off.This unexpected behavior,barely reported in the literature to the

authors knowledge,is caused by the plastic wave front triggered by

the abrupt motion of the impacted side at t ¼ 0.In other words,the

plastic wave propagation at the onset of loading inﬂuences the

necking position,and thus the ductility of the sample as discussed in

theprevious paragraph.Aclear illustrationof theinﬂuenceof necking

position on the sample ductility is reported in Fig.8,where the

transverse displacement of the specimen in the X

2

direction is

depicted as a function of the normalized gauge length.For

_

3

¼ 4000 s

1

and _

3

¼ 5000 s

1

the neck takes place closer to the

clamped end when the SIMT is switched on.Furthermore,for these

loadingrates,theneckismoredevelopedinthecaseof activatedSIMT.

In other words,the maximumvertical displacement is larger in the

case of activated SIMT.For 6000 s

1

< _

3

< 10;000 s

1

the neck takes

place close to the impacted end independently on the martensitic

transformation,leading to larger values of

3

neck

and E

s

if the SIMT is

active.Finallyit is worthnotingthat for strainrates

_

3

a10;000 s

1

the

CIV has been largely exceeded for both,SIMT switched on and SIMT

Fig.7.Illustration of the interplay between initial strain rate _

3

and necking location.

μ

μ

(a)

(b)

Fig.8.Vertical displacement of the gauge length as a function of the normalized gauge

length for simulations in which the SIMT has been alternatively switched on and off.

(a) Strain rate _

3

¼ 4000 s

1

,loading time t ¼ 175.2

m

s.(b) Strain rate _

3

¼ 5000 s

1

,

loading time t ¼ 150.0

m

s.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216 211

switched off simulations and the energy absorbed by the sample

becomes independent of the material description.

4.2.The role of plastic waves

The role of plastic wave propagation on the onset of necking is

evaluated through the comparison of ﬁeld and no-ﬁeld simulations

in which the SIMT is active.This analysis is split into two parts.

Firstly,we analyze the different results obtained fromthe ﬁeld and

no-ﬁeld simulations in full sample description simulations and

their comparison with the single element computations.The

second part addresses speciﬁcally the material response in the ﬁeld

simulations.

e Fig.9 shows the necking strain

3

neck

and the energy absorbed

by the sample E

s

as a function of the loading rate _

3

for ﬁeld and

no-ﬁeld simulations.Within the range of strain rates

_

3

< 3000 s

1

the results obtained using both initial velocity

conditions are practically identical.In both cases the neck is

incepted in the middle of the sample.Within the range

3000 s

1

<

_

3

< 7500 s

1

the difference arises and both

3

neck

and E

s

are signiﬁcantly larger in the case of the ﬁeld simula-

tions.Then,in the case of no-ﬁeld simulations the neck occurs

in the clamped end and in the case of ﬁeld computations the

neck still occurs in the middle of the sample.Next,within the

range 7500 s

1

< _

3

< 10;000 s

1

the trend is inverted and

3

neck

and E

s

are larger in the case of no-ﬁeld calculations.For ﬁeld

simulations,a transition to two evenly spaced necks appears as

strain rate increases,leading to a transient decrease in the

energy required for necking formation,as will be explained

later.In the case of no-ﬁeld simulations,the neck is nucleated

in the impacted end leading to the CIV.It can be stated that as

soon the neck leaves the middle of the sample in the no-ﬁeld

computations,the difference between ﬁeld and no-ﬁeld

simulations sets in.As mentioned earlier,the plastic wave

front generated by the impact in the case of the no-ﬁeld

simulations is responsible for such behavior.Consequently,at

strain rates above 9000 s

1

,the necking strain for the no-ﬁeld

ε

ε

ε

ε

Fig.9.Necking strain

3

neck

and energy absorbed by the sample E

s

as a function of the

loading rate _

3

for SIMT simulations in which the initial velocity ﬁeld has been switched

on and off.

Fig.10.(a) Plastic strain upon the normalized gauge length at 10,000 s

1

for no-ﬁeld and ﬁeld simulations in which the SIMT has been switched on.(b) Contours of plastic strain at

10,000 s

1

for no-ﬁeld and ﬁeld simulations in which the SIMT has been switched on.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216212

calculation keeps lower than that corresponding to the ﬁeld

case.Fig.10-a shows the plastic strain along the normalized

gauge length in the case of _

3

¼ 10;000 s

1

for both initial

velocity conditions and different loading times.It can be

observed that if the velocity ﬁeld is not initialized,the plastic

strain concentrates close to the impacted side from the very

beginning of loading similarly to the solution proposed by Von

Kàrman and Duwez [44].On contrary,if the velocity ﬁeld is

initialized,the plastic strain in the sample at the beginning of

loading is largely constant.This conﬁrms the role of the initial

velocity ﬁeld minimizing the plastic wave propagation.In

addition,the results clearly indicate that the plastic waves,and

therefore the necking location,distort the necking strain of

the material.Figs.11 and 12 show the comparison between

stressestrain curves obtained from the full sample computa-

tions and the single element simulations for ﬁeld and no-ﬁeld

conditions at two different initial strain rates,

_

3

¼ 4000 s

1

and

_

3

¼ 7500 s

1

.It can be observed that the stressestrain

characteristics obtained fromthe no-ﬁeld simulations and the

single element results do not match,especially at low strain

levels.This difference increases with the impact velocity.In

other words,as soon as the neck does not take place in the

middle of the gauge the measured material characteristic in the

no-ﬁeld simulations is not representative of the real material

behavior.This ﬁnding demonstrates that the velocity which

imposes an upper limit to the dynamic tension test for deter-

mining material the actual stressestrain curve of the material is

not the CIV but the velocity which causes the necking to move from

the center of the sample to the clamped end.Beyond this limit,

the displacement recorded at the impacted side and the force

recorded at the clamped side do not correspond to a uniform

state of stress and strain along the specimen,as needed for

a proper interpretation of the experimental measurements.

This observation is relevant for experimental work,and the

results shown here were not previously reported to the best of

the authors knowledge.

e Consider now the specimen response in the ﬁeld simulations.

Froma general point of view,the results shown in Fig.9 reveal

that in the absence of plastic wave propagation,both

3

neck

and

E

s

increase with loading rate e this is not true within

4500 s

1

< _

3

< 7500 s

1

as discussed later.This behavior

seems to be explained by the role played by strain rate _

3

on

necking retardation [19,23,29].As expected,the CIV is avoided

in absence of plastic waves.Moreover,it is worth nothing that

for the lower strain rates considered,up to w7500 s

1

,a single

neck located in the middle of the gauge is formed.Then,this

single neck suddenly gives way to the nucleation of two

(rather) evenly spaced necks along the gauge of the sample,

Fig.10,that are also found at higher strain rates.Following

Rodríguez-Martínez et al.[69] this transition may be related to

material inertia aspects;taking into account that the term

inertia not only covers material density but it also accounts for

the intrinsic effects that sample dimensions,ﬂow stress level

and loading rate all have on necking inception as described

elsewhere [19,23,29].This may explain the nucleation of two

regularly spaced necks assuming that at w7500 s

1

,the elon-

gation of the sample before necking is such that it permits the

σ

ε

σ

ε

(a)

(b)

Fig.11.Comparison between stressestrain curves for simulations using the full sample

description and the single element.SIMT is switched on.Loading rate 4000 s

1

.(a) No-

ﬁeld simulation,(b) ﬁeld simulation.

σ

ε

σ

ε

(a)

(b)

Fig.12.Comparison between stressestrain curves for simulations using the full

sample description and the single element.SIMT is switched on.Loading rate 7500 s

1

.

(a) No-ﬁeld simulation,(b) ﬁeld simulation.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216 213

promotion of a more favorable wavelength mode for the

necking formation.This more favorable wavelength should to

be at least twice shorter than the stretched gauge allowing the

formation of two necks.The transition to a more favorable

wavelength is accompanied by a decrease in the energy

required for necking formation which may explain the tran-

sient decrease inthe necking strainreported in Fig.9 withinthe

range 4500 s

1

<

_

3

< 7500 s

1

.Moreover,it was previously

noted that when the necks are incepted close to the shoulders

of the sample,the development of the plastic deformation gets

limited,which can act as an additional factor that decreases the

necking strain within this range of loading rates.It should be

highlighted that while the transition from the formation of

a single neck to the formation of two necks in the ﬁeld simu-

lations is dictated by material inertia aspects,the necking

location in the no-ﬁeld simulations was dictated by the plastic

wave propagation.Fig.13 illustrates the vertical displacement

of the gauge length as a function of the normalized gauge

length for different ﬁeld simulations.Three selectedstrainrates

are 7500 s

1

,12,500 s

1

and 15,000 s

1

,for which two necks

were nucleated.The loading time has been set in order to

obtain the same maximumvertical displacement for the three

cases considered as indicated in Fig.13.For the lower loading

rate considered,the incepted necks develop in a rather

different manner,Fig.13.It has been detected that one of them

e the one closer to the clamped side e develops ﬁrst in time

may be because of the disturbances resulting fromthe sudden

rise in stress experienced by the sample at the onset of loading

e see Section 4.a.Once the ﬁrst neck is formed an unloading

wave emanates from the necked region slowing down the

growth rate of the secondary neck [70].This explanation ﬁnds

agreement with the original Mott’s theory [71] and the

subsequent developments by Grady and co-workers [72e74].

Moreover it can be observed that the two necks thus formed,

become increasingly symmetrical as the loading rate increases,

i e.as the role played by inertia increases.This behavior agrees

with theoretical and numerical observations reported else-

where for other multiple necking problems [22,27,69].

4.3.The role of SIMT in absence of plastic waves

The role of SIMT in absence of plastic waves on the energy

absorption capability of the material is evaluated using ﬁeld initial

conditions.Fig.14 shows the necking strain

3

neck

and the energy

absorbed by the sample E

s

as a function of the loading rate

_

3

for

simulations in which the SIMT has been activated and deactivated.

It can be observed that now the picture is completely different to

that reported in Fig.6;the relative inﬂuence of the SIMT on the

material response is found to be highly dependent on the presence

or absence of the plastic waves.

It is worth noting that the recorded values of

3

neck

seem to be

rather independent of the martensitic transformation,in fact in

some cases

3

neck

is larger in the case of deactivating the SIMT.The

explanation may be related to the emerging role played by inertia e

the concept of inertia was mentioned in Section 4.a eat high strain

rates [19,29,69].In other words,the stabilizing effect of the

enhanced strain hardening provided by the SIMT seems to be

balanced by the destabilizing effect of its associated increase in ﬂow

stress [23].However,although the necking strain values are quite

similar for both conditions,the computations in which the SIMT is

switched on show larger values of E

s

;it has to be noted that such

difference tends to decrease with the increasing loading rate.In

other words,the SIMT does not provide enhanced ductility to the

material but it is still beneﬁcial in terms of energy absorption due to

the larger material ﬂowstress.This calls for a re-assessment of the

usefulness of the SIMT for enhancing material ductility in high

strain rate applications,for which the material response may be

governed by inertia effects.

5.Conclusions

In this paper the processes of strain localization and necking

formation in AISI 304 steel sheets subjected to dynamic tension

have been investigated using ﬁnite element simulations.Two

different numerical conﬁgurations are addressed:single element

simulations and full specimen geometry simulations.In the full

specimen geometry calculations two different initial conditions

have been applied;no-ﬁeld which is representative of a regular

experimental arrangement and ﬁeld which allows minimizing the

propagation of plastic waves along the longitudinal direction of the

sample.The material behavior is described by a constitutive model

proposed by the authors which includes explicitly the SIMT,thus

characterizing the response of the AISI 304 at high loading rates.

The analysis is focused on the effects of SIMT and plastic wave

propagation on the process of ﬂow localization,which in turn

ε

ε

ε

ε

Fig.14.Necking strain

3

neck

and energy absorbed by the sample E

s

as a function of the

loading rate

_

3

for ﬁeld simulations in which the martensitic transformation has been

switched on and off alternatively.

μ

μ

μ

Fig.13.Vertical displacement of the gauge length as a function of the normalized

gauge length for ﬁeld simulations in which the SIMT has been switched on.Three

different loading rates are illustrated:7500 s

1

,12,500 s

1

and 15,000 s

1

.The loading

time has been set in order to obtain the same maximumvertical displacement for the

three cases considered.

J.A.Rodríguez-Martínez et al./International Journal of Impact Engineering 54 (2013) 206e216214

determines the energy absorption capacity of the material at high

loading rates.

The main conclusions that emerge fromthis work are as follows:

e In presence of plastic waves,and under certain strain rate

conditions,the increased strain hardening provided by the

SIMT process may not delay the onset of necking,thus limiting

the energyabsorption capacity of the material.This unexpected

behavior is barely reported in the literature,and it is related to

the motion of the necking location along the gauge length.

e Once a certain impact velocity is exceeded,the presence of

plastic waves in the dynamic tensile test hinders the actual

material behavior.In other words,as soon as the neck does not

take place in the middle of the gauge,the stressestrain curve

obtained fromdynamic tensile tests is not representative of the

actual material behavior.This ﬁnding demonstrates that the

velocity which imposes an upper limit to the dynamic tension

test for determining material properties is not the CIV,but the

velocity which causes the necking to move fromthe center of

the sample.

e In the absence of plastic waves,and under certain strain rate

conditions,the SIMT may not provide the anticipated enhanced

material ductility.The stabilizing effect of the enhanced strain

hardening provided by the SIMT is balanced by the destabi-

lizing effect of its associated increase in ﬂow stress.This

behavior seems to be a limiting factor to the homogeneous

deformation behavior of the material at high strain rates.This

observation calls for a re-assessment of the beneﬁcial effects of

SIMT at high loading rates,for which the material response

may be governed by inertial effects.

Acknowledgments

J.A.Rodríguez-Martínez and R.Zaera express sincere gratitude

to Dr.Guadalupe Vadillo,Professor José Fernández-Sáez and

Professor Alain Molinari for helpful discussions on the role played

by material aspects on the formation of plastic instabilities in

ductile materials subjected to high strain rates.

D.Rittel acknowledges the support of Carlos III University with

a Cátedra de Excelencia funded by Banco Santander during academic

year 2011e2012.

The researchers of the University Carlos III of Madrid are

indebted to the Comunidad Autónoma de Madrid (Project CCG10-

UC3M/DPI-5596) and to the Ministerio de Ciencia e Innovación de

España (Project DPI/2008-06408) for the ﬁnancial support received

which allowed conducting part of this work.

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